Method for lossless data compression using greedy sequential grammar transform and sequential encoding

ABSTRACT

A method of lossless data compression is provided which uses a grammar transform to sequentially construct a sequence of irreducible context-free grammars from which an original data sequence can be recovered incrementally. The data sequence is encoded using any one of a sequential encoding method, an improved sequential encoding method and a hierarchical encoding method.

This application claims the benefit of provisional U.S. application Ser. No. 60/172,127, filed Dec. 17, 1999.

FIELD OF THE INVENTION

The present invention relates to a method of encoding a data sequence by converting the data sequence into a grammar transform and then losslessly encoding the data sequence based on the grammar transform.

BACKGROUND OF THE INVENTION

Universal data compression methods can be divided into two subdivisions: universal loss less data compression and universal lossy data compression. Conventional universal lossless data compression methods typically employ arithmetic coding algorithms, Lempel-Ziv algorithms, and their variants. Arithmetic coding algorithms and their variants are statistical model-based algorithms. To use an arithmetic coding algorithm to encode a data sequence, a statistical model is either built dynamically during the encoding process, or assumed to exist in advance. Several approaches have been proposed to dynamically build a statistical model. These include the prediction by partial match algorithm, dynamic Markov modeling, context gathering algorithm, and context-tree weighting method. In all of these methods, the next symbol in the data sequence is typically predicted by a proper context and coded by the corresponding estimated conditional probability. Good compression can be achieved if a good trade-off is maintained between the number of contexts and the conditional entropy of the next symbols given contexts during the encoding process. Arithmetic coding algorithms and their variants are universal only with respect to the class of Markov sources with Markov order less than some designed parameter value. Note that in arithmetic coding, the original data sequence is encoded letter by letter. In contrast, no statistical model is used in Lempel-Ziv algorithms and their variants. During the encoding process, the original data sequence is parsed into non-overlapping, variable-length phrases according to a string matching mechanism, and then encoded phrase by phrase. Each parsed phrase is either distinct or replicated with the number of repetitions less than or equal to the size of the source alphabet. Phrases are encoded in terms of their positions in a dictionary or database. Lempel-Ziv algorithms are universal with respect to a class of sources which is broader than the class of Markov sources of bounded order; the incremental parsing Lempel-Ziv algorithm is universal for the class of stationary, ergodic sources. Other conventional universal compression methods include the dynamic Huffman algorithm, the move-to-front coding scheme, and some two stage compression algorithms with codebook transmission. These conventional methods are either inferior to arithmetic coding and Lempel-Ziv algorithms, or too complicated to implement. Recently, J. C. Kieffer and E. H. Yang proposed a class of lossless data compression algorithms based on substitution tables in “Lossless Data Compression Algorithms Based on Substitution Tables”, Proc. of the 1998 Canadian Conference on Electrical and Computer Engineering(Waterloo, Ontario), Vol. 2, pp. 629-632, May 24-28, 1998. In this paper, a new coding framework is presented, but no explicit data compression algorithm is proposed. The greedy sequential transformation discussed in the paper is difficult to implement and does not facilitate subsequent efficient coding, since the symbol s₀ is involved in the parsing process.

SUMMARY OF THE INVENTION

In accordance with the present invention, a universal lossless data compression method is provided to overcome the above-described disadvantages of existing compression methods. The compression method of the present invention utilizes a grammar transform to sequentially construct a sequence of irreducible context-free grammars from which an original data sequence can be recovered incrementally. Based on the grammar transform, the data sequence is encoded using sequential or hierarchical encoding methods.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments of the present invention are described in detail below with reference to the attached drawing figures, in which:

FIG. 1 is a flow chart illustrating a sequence of operations for a grammar transform in accordance with an embodiment of the present invention;

FIGS. 2, 3 and 4 are flow charts illustrating a sequence of operations for a grammar transform in accordance with an embodiment of the present invention;

FIGS. 5, 6 and 7 are flow charts illustrating a sequence of operations for sequential compression in accordance with an embodiment of the present invention;

FIGS. 8, 9, 10, 11 and 12 are flow charts illustrating a sequence of operations for sequential compression in accordance with an embodiment of the present invention; and

FIG. 13 is a flow chart illustrating a sequence of operations for hierarchical compression in accordance with an embodiment of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Prior to describing the methods of the present invention in detail, terms and notation used will now be described. Let A be a source alphabet with cardinality greater than or equal to 2. Let A* be the set of all finite strings drawn from A, including the empty string λ, and A⁺ the set of all finite strings of positive length from A. The notation |A| stands for the cardinality of A, and for any x∈A*, |x| denotes the length of x. For any positive integer n, A^(n) denotes the set of all sequences of length n from A. To avoid possible confusion, a sequence from A is sometimes called an A-sequence. The source alphabet A is of course application-dependent. Without loss of generality, it is assumed that A={0, 1, . . . , |A, A|1}. In most applications, |A|=256. Fix a countable set S={s₀,s₁,s₂, . . . , } of symbols, disjoint from A. (In a typical implementation, one may select S as {|A|, |A|+1, |A|+2, . . . , }.) Symbols in S will be called variables; symbols in A will be called terminal symbols. For any j≧1, let S(j)={s₀,s₁, . . . , s_(j−1)}. A context-free grammar G is a mapping from S(j) to (S(j)∪A)⁺ for some j≧1. The set S(j) shall be called the variable set of G and, to be specific, the elements of S(j) shall be called sometimes G-variables. To describe the mapping explicitly, for each s_(i)(i<j), the relationship (s_(i), G(s_(i))) is written as s_(i)→G(s_(i)), and referred to as a production rule. Thus the grammar G is completely described by the set of production rules {s_(i)→G(s_(i)): 0≦i≦j}. Start with the variable so. By replacing in parallel each variable s in G(s₀) by G(s), another sequence from S(j)∪A is obtained. If this parallel replacement procedure is repeatedly performed, one of the following will hold:

(1) After finitely many parallel replacement steps, a sequence from A is obtained; or

(2) the parallel replacement procedure never ends because each string so obtained contains an entry which is a G-variable.

For the purpose of data compression, the focus is on grammars G for which the parallel replacement procedure terminates after finitely many steps and every G-variable s_(i)(i<j) is replaced at least once by G(s_(i)) in the whole parallel replacement process. Such grammars G are called admissible grammars and the unique sequence from A resulting from the parallel replacement procedure is called a sequence represented by G or by s₀. Since each variable si is replaced at least once by G(s_(i)), it is easy to see that each variable s_(i)(i≠0) represents a substring of the A-sequence represented by s₀, as shown in the following example.

EXAMPLE 1

Let A={0, 1}. Below is an example of an admissible grammar G with variable set {s₀,s₁,s₂,s₃}.

s₀→0s₃s₂s₁s₁s₃10

s₁→01

s₂→s₁1

s₃→s₁s₂

Perform the following parallel replacements:

In the above, start with so and then repeatedly apply the parallel replacement procedure. After four steps,

a sequence from A is obtained and the parallel replacement procedure terminates. Also, each variable s_(i)(0≦i<4) is replaced at least once by G(s_(i)) in the whole parallel replacement process. Therefore in this example, s₀ (or G) represents the sequence x=00101101101010101110. Each of the other G-variables represents a substring of x: s₁ represents 01, s₂ represents 011, and s₃ represents 01011.

An admissible grammar G is said to be irreducible if the following conditions are satisfied

(1) Each G-variable s other than s₀ appears at least twice in the range of G.

(2) There is no non-overlapping repeated pattern of length greater than or equal to 2 in the range of G.

(3) Each distinct G-variable represents a distinct A-sequence.

The admissible grammar shown in Example 1 is irreducible.

A grammar transform is a transformation that converts any data sequence x∈A⁺ into an admissible grammar G_(x) that represents x. A grammar transform is said to be irreduccible if G_(x) is irreducible for all x∈A⁺.

Starting with an admissible grammar G that represents x, a set of reduction rules can be repeatedly applied to generate another admissible grammar G′ which represents the same x and satisfies Properties (1) to (3) above. This set of reduction rules is described below.

Reduction Rule 1. Let s be a variable of an admissible grammar G that appears only once in the range of G. Let s′ X ceso be the unique production rule in which s appears on the right. Let s→γ be the production rule corresponding to s. Reduce G to the admissible grammar G′ obtained by removing the production rule s→γ from G and replacing the production rule s′→αsβ with the production rule s′→αγβ. The resulting admissible grammar G′ represents the same sequence x as does G.

EXAMPLE 2

Consider the grammar G with variable set {s₀,s₁,s₂} given by {s₀→s₁s₁, s₁→s₂1, s₂→010}. Applying Reduction Rule 1, one gets the grammar G′ with variable set {s₀,s₁} given by {s₀→s₁s₁, s₁→0101}.

Reduction Rule 2. Let G be an admissible grammar possessing a production rule of form s→α₁βα₂βα₃, where the length of β is at least 2. Let s′∈S be a variable which is not a G-variable. Reduce G to the grammar G′ obtained by replacing the production rule s→α₁βα₂βα₃ of G with s→α₁s′α₂s′α₃, and by appending the production rule s′→β. The resulting grammar G′ includes a new variable s′ and represents the same sequence x as does G.

EXAMPLE 3

Consider the grammar G with variable set {s₀,s₁} given by {s₀→s₁01s₁01, s₁→11}. Applying Reduction Rule 2, one gets the grammar G′ with variable set {s₀,s₁,s₂} given by {s₀→s₁s₂s₁s₂, s₁→11, s₂→01}.

Reduction Rule 3. Let G be an admissible grammar possessing two distinct production rules of form s→α₁βα₂ and s′→α₃βα₄, where β is of length at least two, either α₁ or α₂ is not empty, and either α₃ or α₄ is not empty. Let s″∈S be a variable which is not a G-variable. Reduce G to the grammar G′ obtained by doing the following: Replace rule s→α₁βα₂ by s→α₁s″α₂, replace rule s′→α₃βα₄ by s′→α₃βα₄, and append the new rule s″→β.

EXAMPLE 4

Consider the grammar G with variable set {s₀,s₁,s₂} given by {s₀→s₁0s₂, s₁→10, s₂→0s₁0}. Applying Reduction Rule 3, one gets the grammar G′ with variable set {s₀,s₁,s₂,s₃} given by {s₀→s₃s₂, s₁→10, s₂→0s₃, s₃→s₁0}.

Reduction Rule 4. Let G be an admissible grammar possessing two distinct production rules of the form s→α₁βα₂ and s′→β, where β is of length at least two, and either α₁ or α₂ is not empty. Reduce G to the grammar G′ obtained by replacing the production rule s→α₁βα₂ with the production rule s→α₁s′α₂.

EXAMPLE 5

Consider the grammar G with variable set {s₀,s₁,s₂} given by {s₀→s₂01s₁, s₁→s₂0, s₂→11}. Applying Reduction Rule 4, one gets the grammar G′ with variable set {s₀,s₁,s₂} given by {s₀→s₁1s₁, s₁→s₂0, s₂→11}.

Reduction Rule 5. Let G be an admissible grammar in which two variables s and s′ represent the same substring of the A-sequence represented by G. Reduce G to the grammar G′ obtained by replacing each appearance of s′ in the range of G by s and deleting the production rule corresponding to s′. The grammar G′ may not be admissible since some G′-variables may not be involved in the whole parallel replacement process of G′. If so, further reduce G′ to the admissible grammar G″ obtained by deleting all production rules corresponding to variables of G′ that are not involved in the whole parallel replacement process of G′. Both G and G″ represent the same sequence from A.

Embodiment 1

The grammar transform according to a preferred embodiment will now be described. Let x=x₁x₂ . . . x_(n) be a sequence from A which is to be compressed. The irreducible grammar transform is a greedy one in that it parses the sequence x sequentially into non-overlapping substrings {x₁,x₂ . . . x_(n) ₂ , . . . , x_(n) _(t−1) ₊₁ . . . x_(n) _(t) } and builds sequentially an irreducible grammar G_(i) for each x₁ . . . x_(n) _(i) , where 1≦i≦t, n₁=1, and n_(t)=n. The first substring is x₁ and the corresponding irreducible grammar G₁ consists of only one production rule s₀→x₁. Suppose that x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(i−1) ₊₁ . . . x_(n) _(i) have been parsed off and the corresponding irreducible grammar G_(i) for x₁ . . . x_(n) _(i) has been built. Suppose that the variable set of G_(i) is equal to S(j_(i))={s₀,s₁, . . . , s_(j) _(i) ⁻¹}, where j₁=1. The next substring x_(n) _(i) ₊₁ . . . x_(n) _(i+1) is the longest prefix of x_(n) _(i) ₊₁ . . . x_(n) that can be represented by s_(j) for some 0<j<j_(i) if such a prefix exists. Otherwise, x_(n) _(i) ₊₁ . . . x_(n) _(i+1) =x_(n) _(i) ₊₁ with n_(i+1)=n_(i)+1. If n_(i+1)−n_(i)>1 and x_(n) _(i) ₊₁ . . . x_(n) _(i+1) is represented by s_(j), then append s_(j) to the right end of G_(i)(s₀); otherwise, append the symbol x_(n) _(i) ₊₁ to the right end of G_(i)(s₀). The resulting grammar is admissible, but not necessarily irreducible. Apply Reduction Rules 1-5 to reduce the grammar to an irreducible grammar G_(i+1). Then G_(i+1) represents x₁ . . . x_(n) _(i+1) . Repeat this procedure until the whole sequence x is processed. Then the final irreducible grammar G_(t) represents x.

FIG. 1 illustrates the functional blocks of the proposed grammar transform. Using parameters S(j_(i))={s₀, s₁, . . . , s_(j) _(i) ⁻¹} and n_(i) indicated at 7 and 8, the parser 5 parses off the next phrase x_(n) _(i) _(+1 . . . x) _(n) _(i+1) and then outputs the symbol β_(i+1)∈A∪(S(j_(i))−{s₀}) which represents the phase x_(n) _(i) ₊₁ . . . x_(n) _(i+1) . After receiving β_(i+1), the grammar updation operation 6 reduces the appended grammar G_(i) into the irreducible grammar G_(i+1) and then sends the information regarding S(j_(i+1)) and n_(i+1) back to the parser. The following example shows how the proposed grammar transform works.

EXAMPLE 6

Let A={0, 1} and x=10011100010001110001111111000. Apply the above irreducible grammar transform to x. It is easy to see that the first three parsed substrings(or phrases) are 1, 0, and 0. The corresponding irreducible grammars G₁, G₂, and G₃ are given by {s₀→1}, {s₀→10}, and {s₀→100}, respectively. Since j₃=1, the fourth parsed phrase is x₄=1. Appending the symbol 1 to the end of G₃(s₀) gives rise to an admissible grammar G′₃ given by {s₀→1001}. G′₃ itself is irreducible; so none of Reduction Rules 1 to 5 can be applied and G₄ is equal to G′₃. Similarly, the fifth and sixth parsed phrases are x₅=1 and x₆=1, respectively; G₅ and G₆ are given respectively by {s₀→10011} and {s₀→100111}. The seventh parsed phrase is x₇=0. Appending the symbol 0 to the end of G₆(s₀) yields an admissible grammar G′₆ given by

s₀→1001110.

G′₆ is not irreducible any more since there is a non-overlapping repeated pattern 10 in the range of G′₆. At this point, only Reduction Rule 2 is applicable. Applying Reduction Rule 2 once gives the irreducible grammar G₇ given by

s₀→s₁011s₁

s₁→10.

Since the sequence from A represented by s₁ is not a prefix of the remaining part of x, the next parsed phrase is x₈=0. Appending the symbol 0 to the end of G₇(s₀) yields an admissible grammar G′₇ given by

s₀→s₁011s₁0

s₁→10.

G′₇ is not irreducible. Applying Reduction Rule 2 once, which is the only applicable reduction rule at this point, gives rise to a grammar G″₇

s₀→s₂11s₂

s₁→10

s₂→s₁0.

In the above, the variable s₁ appears only once in the range of G″₇. Applying Reduction Rule 1 once gives the irreducible grammar G₈:

s₀→s₁11s₁

s₁→100.

From G₇ to G₈, Reduction Rule 2 followed by Reduction Rule 1 has been applied. Based on G₈, the next two parsed phrases are x₉=0 and x₁₀x₁₁x₁₂=100, respectively. The irreducible grammar G₉ is given by

s₀→s₁11s₁0

s₁→100,

and the grammar G₁₀ is given by

s₀→s₁11s₁0s₁

s₁→100.

Note that from G₉ to G₁₀, the symbol s₁ is appended to the end of G₉(s₀) since the phrase x₁₀x₁₁x₁₂ is represented by s₁. The eleventh parsed phrase is x₁₃=0. Appending 0 to the end of G₁₀(s₀) and then applying Reduction Rule 2 once yield G₁₁

s₀→s₁11s₂s₂

s₁→100

s₂→s₁0.

The twelfth parsed phrase is x₁₄=1 and G₁₂ is obtained by simply appending 1 to the end of G₁₁(s₀). The thirteenth parsed phrase is x₁₅=1. Appending 1 to the end of G₁₂(s₀) and then applying Reduction rule 2 once give rise to G₁₃

s₀→s₁s₃s₂s₂s₃

s₁→100

s₂→s₁0

s₃→11.

The fourteenth parsed phrase is x₁₆x₁₇x₁₈x₁₉=1000, which is represented by s₂. Appending s₂ to the end of G₁₃(s₀) and then applying Reduction Rule 2 followed by Reduction Rule 1 gives rise to G₁₄

s₀→s₁s₃s₂s₃

s₁→100

s₂→s₁0

s₃→11s₂.

The fifteenth parsed phrase is x₂₀=1, and G₁₅ is obtained by appending 1 to the end of G₁₄(s₀). The sixteenth parsed phrase is x₂₁=1. Appending 1 to the end of G₁₅(s₀) and then applying Reduction Rule 3 once result in G₁₆

s₀→s₁s₃s₂s₃s₄

s₁→100

s₂→s₁0

s₃→s₄s₂

s₄→11.

The seventeenth parsed phrase is x₂₂x₂₃=11 and G₁₇ is obtained by appending s₄ to the end of G₁₆(s₀). The final parsed phrase is x₂₄ . . . x₂₉=111000 and G₁₈ is obtained by appending s₃ to the end of G₁₇(s₀). In summary, the proposed irreducible grammar transform parses x into {1, 0, 0, 1, 1, 1, 0, 0, 0, 100, 0, 1, 1, 1000, 1, 1, 11, 111000} and transforms x into the irreducible grammar G₁₈

s₀→s₁s₃s₂s₃s₄s₄s₃

s₁→100

s₂→s₁0

s₃→s₄s₂

s₄→11.

It is apparent from Example 6, to get G_(i+1) from the appended G_(i), Reduction Rules 1 to 3 are employed. Furthermore, the order by which these rules are applied is unique, and the number of times these rules need to be applied is at most 2. This phenomenon is true not only for Example 6, but also for all other sequences.

FIGS. 2, 3 and 4 are flow diagrams illustrating the implementation of the grammar transform in accordance with a preferred embodiment of the present invention. Let G′_(i) denote the appended G_(i). In view of the above, there are two major operations in the grammar transform: the parsing of x into non-overlapping substrings {x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(t−1) ₊₁ . . . x_(n) _(t) } and the updating of G_(i) into G_(i+1) through G′_(i), i=1, 2, . . . , t−1. Before presenting the implementation details of these operations, let us first describe how to represent G_(i).

To represent each G_(i), two dynamic two dimensional arrays are allocated: a symbol array D1 and a string array D2. Let D1(s_(j)) and D2(s_(j)) denote the rows of D1 and D2 corresponding to s_(j), respectively, where 0≦j≦j_(i)−1. The row D1 (s_(j)) is used to retain the information regarding G_(i)(s_(j)), and the row D2(s_(j)) is used to retain the A-string represented by s_(j). Since A is assumed to be {0, 1, . . . , |A|−1}, the variable s_(j) can be represented conveniently by the integer j+|A|. The row D1(s_(j)) consists of symbols from {−1}∪A∪{s₀, s₁, . . . }. The row D1 (s_(j)) itself is not equal to G_(i)(s_(j)); however, with the removal of the special symbol −1 from D1(s_(j)), the modified D1(s_(j)) is then exactly the same as G_(i)(s_(j)). The introduction of the special symbol −1 enables us to update G_(i) into G_(i+1) by locally changing several elements in the two dimensional symbol array D1, thus reducing the computational time spent in the process of updating G_(i) into G_(i+1). Conceptually, the special symbol −1 can be considered as the empty string.

The implementation of the parsing operation is illustrated in Blocks 10 to 13 of FIG. 2. Initially, for any γ∈A, D2(γ) denotes the letter γ itself. Given the irreducible grammar G_(i) for x₁ . . . x_(n) _(i) , the next parsed phrase x_(n) _(i) ₊₁ . . . x_(n) _(i+1) is the longest prefix of the remaining sequence x_(n) _(i) ₊₁ . . . x_(n) that is equal to D2(β) for some β∈A∪(S(j_(i))−{s₀}). In other words, to parse off the next phrase, one simply needs to find β∈A∪(S(j_(i))−{s₀}) such that D2(β) is the longest prefix of the remaining sequence over A∪(S(j_(i))−{s₀}). Store all strings D2(γ), γ∈A∪(S(j_(i))−{s₀}), into a trie structure. Then one can find βby simply using the remaining sequence to traverse the trie structure.

The implementation of the updating operation is illustrated in Blocks 14 to 17 of FIG. 2, in FIG. 3, and in FIG. 4. To facilitate the updating process, notions of an index function and lists are introduced. An index function I: {1, 2, . . . , t}→{0, 1} is defined as follows: I(1)=0, and for any i>1, I(i) is equal to 0 if G_(i) is equal to the appended G_(i−1), and 1 otherwise. In flow diagrams shown in FIGS. 2 through 4, this index function is represented by the symbol I which is initialized to be 0 in Block 10 of FIG. 2. Let α be the last integer in D1(s₀); α is furnished in Block 14 of FIG. 2, where n1 represents the total length of all previously parsed substrings. Let β be the symbol from A∪(S(j_(i))−{s₀}) that represents the next parsed phrase x_(n) _(i) ₊₁ . . . x_(n) _(i+1) ; β is furnished in Block 13 of FIG. 2. The pattern αβ is the only possible non-overlapping repeated pattern of length ≧2 in G′_(i). Furthermore, if αβ is a non-overlapping repeated pattern in the range of G′_(i), then αβ repeats itself only once in the range of G′_(i) since G_(i) is irreducible. To find out whether or not αβ is a non-overlapping repeated pattern and, if it is, where it repeats itself, two lists L₁(γ) and L₂(γ) are allocated to each symbol γ∈{0, 1, . . . |A|−1, |A|+1, . . . , j_(t)+|A|−1}. The list L₁(γ) consists of all vectors (η, s_(m), n), where η∈A∪{s₁, . . . , s_(j) _(i) ⁻¹}, and n is an integer, such that

(a.1) γ is the nth element in D1(s_(m)).

(a.2) η is the first element in D1(s_(m)) which appears after the nth position and is riot equal to −1.

(a.3) When s_(m)=s₀, i.e., when m=0, η does niot locate in the last position of D1(s₀).

(a.4) With the removal of all possible −1 from D1(s_(m)), the modified D1(s_(m)) is not equal to γη.

(a.5) When γ=η and when there is a pattern γγγ appearing in the modified D1(s_(m)), n is not equal to the position of the first γ of the pattern in D1(s_(m)).

The list L₂(γ) consists of all vectors (η, s_(m), n) such that Properties (a.1) to (a.3) and Property (a.5) hold. Only lists L₁(γ) are necessary to perform grammar updation. However, the improved sequential compression algorithm described later uses both L₁(γ) and L₂(γ) to perform encoding. Elements in the lists L₁(γ) and L₂(γ) can be arranged in increasing order of η, or can be stored in trie structures. To facilitate discussion, lists L₁(γ) and L₂(γ) may also be referred to as the sets consisting of all the first components of elements in L₁(γ) and L₂(γ), respectively. Initially, all lists are empty, as shown in Block 10 of FIG. 2. Since G_(i) is uniquely specified by the two dynamic two dimensional arrays D1 and D2, updating G_(i) into G_(i+1) is now equivalent to updating the two dynamic arrays D1 and D2. With the help of the lists L₁(γ), γ∈{0, 1, |A|−1, |A|+1, . . . , j_(t)+|A|−1}, the updating operation can now be implemented as follows.

1 After parsing the substring β from the previously unparsed symbols of the data sequence x, as shown in Blocks 10 to 14 of FIG. 2, it is determined whether or not β appears in the list L₁ (α) as the first component of some element in L₁(α). This step is illustrated in Block 15 of FIG. 2.

2 As shown in Blocks 17 to 21 of FIGS. 2 and 3, if β does not appear as the first component of some element in L₁(α), simply append β to the end of D1(s₀), and update the lists L₁(γ) and L₂(γ) by inserting an additional vector (α,s₀, n), where γ locates in the nth position of the unappended D1(s₀) and is the immediate symbol to the left of the end of the unappended D1(s₀), which is from A∪(S(j_(i))−{s₀}). In this case, G_(i+1) is equal to G′_(i), and no new variable is created.

3 If β appears as the first component of some element in L₁(α) and if I(i)=0 (Block 24), then only Reduction Rule 2 or 3 is applicable. Suppose that (β, s_(m), k) is an element in L₁(α). The vector (β, s_(m), k) is furnished in Block 22 of FIG. 4. Then the pattern αβ repeats itself only in the kth position of D1(s_(m)), with β occurring, say, in the lth position of D1(s_(m)). Any integer occurring between the kth position and the lth position in D1(s_(m)), if any, is equal to −1. The integer l is computed in Block 23 of FIG. 4. Change the symbol β in the lth position of D1(s_(m)) to −1, and the symbol α in the kth position of D1(s_(m)) to the new variable s_(j) _(i) , which is equal to |A|+j_(i). Change the last symbol in D1(s₀), which is α, to be s_(j) _(i) . Create a new row D1(s_(j) _(i) ) of D1 to denote the pattern αβ. Update the lists L₁(γ) and L₂(γ) accordingly (at most, eight lists need to be updated.) Create a new row D2(s_(j) _(i) ) of D2 to retain the A-string represented by the new variable s_(j) _(i) . These steps are illustrated in Blocks 25 to 30 of FIG. 4.

4 If β appears as the first component of some element in L₁(α) and if I(i)=1(Block36), then Reduction Rule 2 or 3 followed by Reduction Rule 1 is applicable. In this case, α is the newly created variable s_(j) _(i) ⁻¹, and the list L₁(α) contains only one vector (β, s_(m), k). The pattern αβ repeats itself only in the kth position of D1(s_(m)) with β occurring, say, in the lth position of the row D1(s_(m)). Change the integer β in the lth position of D1(s_(m)) to −1, and append β to the end of the row D1(α). Update L₁(α), L₁(β), and other lists accordingly. (At most, seven lists need to be updated.) Append the A-string represented by β to the end of the row D2(α). These steps are illustrated in Blocks 31 to 35 of FIG. 4.

Embodiment 2

In another embodiment of the present invention, the data sequence x is encoded sequentially by using a zero order arithmetic code with a dynamic alphabet to encode the sequence of parsed phrases x₁,x₂ . . . x_(n) ₂ , . . . , x_(n) _(t−1) ₊₁ . . . x_(n) _(t) . The resulting algorithm is called a sequential algorithm. Specifically, each symbol β∈S∪A is associated with a counter c(β). Initially, c(β) is set to 1 if β∈A and 0 otherwise. At the beginning, the alphabet used by the arithmetic code is A. The first parsed phrase x₁ is encoded by using the probability c(x₁)/Σ_(β∈A)c(β). Then the counter c(x₁) increases by 1. Suppose that x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(i−1) ₊₁ . . . x_(n) _(i) have been parsed off and encoded and that all corresponding counters have been updated. Let G_(i) be the corresponding irreducible grammarfor x₁ . . . x_(n) _(i) . Assume that the variable set of G_(i) is equal to S(j_(i))={s₀, s₁, . . . , s_(j) _(i) ⁻¹}. Let x_(n) _(i) ₊₁ . . . x_(n) _(i+1) be parsed off as in the irreducible grammar transform of the present invention and represented by β∈{s₁, . . . , s_(j) _(i) ⁻¹}∪A. Encode x_(n) _(i) ₊₁ . . . x_(n) _(i+1) (or β) and update the relevant counters according to the following steps:

Step 1: The alphabet used at this point by the arithmetic code is {s₁, . . . , s_(j) _(i) ⁻¹}∪A. Encode x_(n) _(i) ₊₁ . . . x_(n) _(i+1) by using the probability $\begin{matrix} {{c(\beta)}/{\sum\limits_{\alpha \in {{S{(j_{i})}}\bigcup A}}^{\quad}{{c(\alpha)}.}}} & (1) \end{matrix}$

Step 2: Increase c(β) by 1.

Step 3: Get G_(i+1) from the appended G_(i) as in our irreducible grammar transform.

Step 4: If j_(i+1)>j_(i), i. e. , G_(i+1) includes the new variable s_(j) _(i) , increase the counter c(s_(j) _(i) ) by 1.

Repeat this procedure until the whole sequence x is processed and encoded.

EXAMPLE 7

The sequential algorithm is applied to compress the sequence x=10011100010001110001111111000 shown in Example 6. It follows from Example 6 that x is parsed into {1, 0, 0, 1, 1, 1, 0, 0, 0, 100, 0, 1, 1, 1000, 1, 1, 11, 111000}. The product of the probabilities used to encode these parsed phrases is $p = {\frac{1}{2}\frac{1}{3}\frac{2}{4}\frac{2}{5}\frac{3}{6}\frac{4}{7}\frac{3}{8}\frac{4}{10}\frac{5}{11}\frac{1}{12}\frac{6}{13}\frac{5}{15}\frac{6}{16}\frac{1}{18}\frac{7}{19}\frac{8}{20}\frac{1}{22}{\frac{1}{23}.}}$

FIGS. 5, 6 and 7 list the functional flow diagrams of the proposed sequential compression algorithm. The parsing, encoding, and updating operations are interleaved. Block 37 of FIG. 5 is essentially the same as Block 10 of FIG. 2; the only difference is that Block 37 contains an additional parameter “deno”, which is designated to be the denominator of the fraction number shown in (1) and initialized to be |A|. Blocks 40 to 42, 44, 45, and 47 of FIG. 5 are the same as Blocks 11 to 15, and 17 of FIG. 2, respectively. All blocks of FIG. 6 except Block 56 are the same as those of FIG. 4. Block 56 implements Step 4 above, and also increases the denominator “deno” by 1. Blocks 64 to 66 of FIG. 7 implement Steps 1 and 2 above (i.e., Blocks 38, 39 and 43 of FIG. 5).

Embodiment 3

In a third embodiment of the present invention, the index sequence {I(i)}_(i=1) ^(t) and the structures of {G_(i)}_(i=1) ^(t) are used to improve the encoding of the sequence of parsed phrases x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(t−1) ₊₁ . . . x_(n) _(t) . The resulting algorithm is called an improved sequential algorithm. The sequence {I(i)}_(i=1) ^(t) itself is encode by an order 1 arithmetic code. In addition to the counters c(γ), γ∈S∪A, new counters c(0, 0), c(0, 1), c(1, 0), c(1, 1) and ĉ(γ) are defined. The counters c(0, 0), c(0, 1), c(1, 0), and c(1, 1) are used to encode the sequence {I(i)}_(i=1) ^(t); initially, they are all equal to 1. The (i+1)th parsed phrase is encoded by the counters ĉ(γ) whenever I(i)=0 and I(i+1)=1 and by the counters c(γ) whenever I(i+1)=0. As in the case of c(γ), initially ĉ(γ) is 1 if γ∈A and 0 if γ∈S. The first three parsed phrases are encoded as in the sequential algorithmrr since they are x₁, x₂, and x₃. Also, I(1), I(2), and I(3) are all 0 and hence there is no need to encode them. Starting from the fourth phrase, we first encode I(i+1), and then use I(i+1) as side information and the structure of G_(i) as context information to encode the (i+1)th parsed phrase. Suppose that x₁, x₂ . . . x_(n) ₂ . . . x_(n) _(i−1) ₊₁ . . . x_(n) _(i) and I(4), . . . , I(i) have been encoded and that all corresponding counters have been updated. Let G_(i) be the corresponding irreducible grammar for x₁ . . . x_(n) _(i) . Assume that the variable set of G_(i) is equal to S(j_(i))={s₀, s₁, . . . , S_(j) _(i) ⁻¹}. Let ae be the last symbol of G_(i)(s₀). Let x_(n) _(i) ₊₁ . . . x_(n) _(i+1) be parsed off as in our irreducible grammar transform and represented by β∈{s₁, . . . , s_(j) _(i) ⁻¹}∪A. Encode I(i+1) and β, and update the relevant counters according to the following steps:

Step 1: Encode I(i+1) by using the probability

c(I(i),I(i+1))/(c(I(i),0)+c(I(i),1)).

Step 2: Increase c(I(i), I(i+1)) by 1.

Step 3: If I(i+1)=0, encode β by using the probability $\begin{matrix} {{{c(\beta)}/{\sum\limits_{\gamma \in {{S{(j_{i})}}\bigcup{A - {L_{2}{(\alpha)}}}}}^{\quad}{c(\gamma)}}},} & (2) \end{matrix}$

and then increase c(β) by 1. If I(i)=0 and I(i+1)=1, encode β by using the probability $\begin{matrix} {{{\hat{c}(\beta)}/{\sum\limits_{\gamma \in {L_{1}{(\alpha)}}}^{\quad}{\hat{c}(\gamma)}}},} & (3) \end{matrix}$

and then increase ĉ(β) by 1. On the other hand, if I(i)=1 and I(i+1)=1, no information is sent since L₁(α) contains only one element and the decoder knows what β is.

Step 4: Get G_(i+1) from the appended G_(i) as in our irreducible grammar transform. Update L₁(γ) and L₂(γ) accordingly, where γ∈S(j_(i+1))∪A.

Step 5: If j_(i+1)>j_(i), i.e., G_(i+1) includes the new variable s_(j) _(i) , increase both c(s_(j) _(i) ) and ĉ(s_(j) _(i) ) by 1.

Repeat this procedure until the whole sequence x is processed and encoded.

In Step 3 above, L₁(γ) and L₂(γ) are considered as the sets consisting of all the first components of elements in L₁(γ) and L₂(γ), respectively.

EXAMPLE 7

The improved sequential algorithm is applied to compress the sequence x=10011100010001110001111111000 shown in Example 6. It follows from Example 6 that x is parsed into {1, 0, 0, 1, 1, 1, 0, 0, 0, 100, 0, 1, 1, 1000, 1, 1, 11, 111000}. The corresponding sequence {I(i)} is 000000110010110100. The product of the probabilities used to encode the sequence {I(i)}_(i=4) ¹⁸ is $\frac{1}{2}\frac{2}{3}\frac{3}{4}\frac{1}{5}\frac{1}{2}\frac{1}{3}\frac{4}{6}\frac{2}{7}\frac{2}{4}\frac{3}{8}\frac{2}{5}\frac{3}{6}\frac{4}{9}\frac{4}{7}{\frac{5}{10}.}$

The product of the probabilities used to encode the parsed phrases is $\frac{1}{2}\frac{1}{3}\frac{2}{4}\frac{2}{5}\frac{3}{3}\frac{4}{4}\frac{1}{2}\frac{3}{4}\frac{1}{6}\frac{2}{3}\frac{5}{12}\frac{1}{5}\frac{6}{13}\frac{2}{5}\frac{1}{15}{\frac{1}{16}.}$

Note that the (i+1)th parsed phrase need not be encoded whenever I(i)=1 and I(i+1)=1.

FIGS. 8 through 12 list the functional flow diagrams of the improved sequential compression algorithm. Once again, the parsing, encoding, and updating operations are interleaved. Block 67 of FIG. 8 initializes all parameters which, in addition to the parameters listed in Block 37 of FIG. 5, includes new parameters c(0, 0), c(0, 1), c(1, 0), c(1, 1), and ĉ(γ). Blocks 70 to 74 of FIG. 8 are the same as Blocks 11 to 15 of FIG. 2. Blocks 77 and 78 of FIG. 9 implement Steps 1 and 2 above in the case of I(i+1)=0. Blocks 79, and 84 to 86 of FIGS. 9 and 10 encode the parsed phrase β in the case of I(i+1)=0. Blocks 80 to 83 of FIG. 9 are the same as those of FIG. 3. All blocks in FIG. 11 except Blocks 89, 90, 92, and 98 are the same as those of FIG. 4. Blocks 89 and 90 implement Steps 1 and 2 above in the case of I(i+1)=1. Blocks 92, 106 to 108, and 98 encode the parsed phrase β in the case of I(i)=0 and I(i+1)=1.

Embodiment 4

In a fourth embodiment of the present invention, the data sequence x is encoded using a hierarchical encoding method. The resulting algorithm is called a hierarchical compression algorithm. Let x be an A sequence which is to be compressed. Let G_(t) be the final irreducible grammar for x furnished by our irreducible grammar transform. In the hierarchical algorithm, a zero order arithmetic code with a dynamic alphabet is used to encode G_(t)(or its equivalent form). After receiving the binary codeword, the decoder recovers G_(t)(or its equivalent form) and then performs the parallel replacement procedure mentioned before to get x.

To illustrate how to encode the final irreducible grammar G_(t), the following example is provided in which t=18. With reference to Block 109 of FIG. 13, G₁₈ is furnished by the proposed irreducible grammar transform and given by

s₀→s₁s₃s₂s₃s₄s₄s₃

s₁→100

s₂→s₁0

s₃→s₄s₂

s₄→11.

The above form of G₁₈, however, is not convenient for transmission. To encode G₁₈ efficiently, G₁₈ is converted into its canonical form G₁₈ ^(g) given by

s₀→s₁s₂s₃s₂s₄s₄s₂

s₁→100

s₂→s₄s₃

s₃→s₁0

s₄→11.

To get G₁₈ ^(g) from G₁₈, the variable s₃ in G₁₈ is renamed as s₂ in G₁₈ ^(g) and the variable s₂ in G₁₈ is renamed as s₃ in G₁₈ ^(g). The difference between G₁₈ ^(g) and G₁₈ is that the following property holds for G₁₈ ^(g), but not for G₁₈:

(c.1) If G₁₈ ^(g)(s_(i)) is read from left to right and from top(i=0) to bottom(i=4), then for any j≧1, the first appearance of s_(j) always precedes that of s_(j+1).

In G₁₈, the first appearance of s₃ precedes that of s₂; this is why these two variables need be renamed to get G₁₈ ^(g). Note that both G₁₈ and G₁₈ ^(g) represent the same sequence x. It is now more convenient to encode and transmit G₁₈ ^(g) instead of G₁₈. To do s_(o), concatenate G₁₈ ^(g) (s₀), G₁₈ ^(g)(s₁), . . . , G₁₈ ^(g)(s₄) in the indicated order, insert the symbol e at the end of G₁₈ ^(g)(s₀) and for any i≧1 satisfying |G₁₈ ^(g)(s_(i))|>2, insert the symbol b at the beginning of G₁₈ ^(g)(s_(i)) and the symbol e at the end of G₁₈ ^(g)(s_(i)), where symbols b and e are assumed not to belong to S∪A. This gives rise to the following sequence from the alphabet S∪A∪{b, e}:

s₁s₂s₃s₂s₄s₄s₂eb100es₄s₃s₁011  (4)

where A {0, 1} in this example. From the sequence given by (4), G₁₈ ^(g) can again be obtained. First, G₁₈ ^(g)(s₀) can be identified by looking at the first appearance of symbol e. Second, all G₁₈ ^(g)(s_(i)) with |G₁₈ ^(g)(s_(i))|>2 can be identified by looking at pairs (b,e). Finally, all the other G₁₈ ^(g)(s_(i)) have length 2. A question may arise as to why both symbols e and b are introduced. After all, e can be inserted at the end of each G₁₈ ^(g)(s_(i)) to identify G₁₈ ^(g)(s_(i)). The reason is that most G_(t)(s_(i)) of any G_(t) furnished by the irreducible grammar transform of the present invention have length 2. As a result, by using the pair (b,e) to isolate G_(t)(s_(i)) with |G_(t)(s_(i))|>2, a shorter concatenated sequence is obtained and therefore improved compression performance. Since G₁₈ ^(g) is canonical, i. e. , G₁₈ ^(g) satisfies Property (c.1), the first appearance of s_(i), for any i≧1, precedes that of s_(i+1) in the sequence given by (4). To take advantage of this in order to get better compression performance, a further step is taken. Let s be a symbol which is not in S∪A∪{b,e}. For each i≧1, replace the first appearance of s_(i) in the sequence given by (4) by s. Then the following sequence from S∪A∪{b,e,s} is obtained:

ssss₂ss₄s₂eb100es₄s₃s₁011  (5)

which is referred to as the sequence generated from G₁₈ or its canonical form G₁₈ ^(g). As evident from the sequence given by (5), the sequence given by (4)is obtained again by simply replacing the ith s in (5) by s_(i). Therefore, from the sequence generated from G₁₈, G₁₈ ^(g) is obtained and hence x. To compress G₁₈ ^(g) or x, a zero order arithmetic code with a dynamic alphabet is used to encode the sequence generated from G₁₈. Specifically, each symbol β∈S∪A∪{b,e,s} is associated with a counter c(β). Initially, c(β) is set to 1 if β∈A∪{b,e,s} and 0 otherwise. The initial alphabet used by the arithmetic code is A∪{b,e,s}. Encode each symbol β in the sequence generated from G₁₈ and update the related counters according to the following steps:

Step 1: Encode β by using the probability ${c(\beta)}/{\sum\limits_{\alpha}^{\quad}{c(\alpha)}}$

where the summation Σ_(α) is taken over A∪{b,e,s}∪{s₁, . . . , s_(i)} and i is the number of times that s occurs before the position of this β. Note that the alphabet used at this point by the arithmetic code is A∪{b,e,s}∪{s₁, . . . , s_(i)}.

Step 2: Increase the counter c(β) by 1.

Step 3: If β=s, increase the counter c(s_(i+1)) from 0 to 1, where i is defined in Step 1.

Repeat the above procedure until the whole generated sequence is encoded. For the generated sequence given by (5), the product of the probabilities used in the arithmetic coding process is $p = {\frac{1}{5}\frac{2}{7}\frac{3}{9}\frac{1}{11}\frac{4}{12}\frac{1}{14}\frac{2}{15}\frac{1}{16}\frac{1}{17}\frac{1}{18}\frac{1}{19}\frac{2}{20}\frac{2}{21}\frac{2}{22}\frac{1}{23}\frac{1}{24}\frac{3}{25}\frac{2}{26}{\frac{3}{27}.}}$

In general, to encode the final irreducible grammar G_(t), the following procedure is applied with reference to FIG. 13: first convert it into its canonical form G_(t) ^(g) (Block 110), then construct the sequence generated from G_(t) (Block 111), and finally use a zero order arithmetic code with a dynamic alphabet to encode the generated sequence.

It should be pointed out that in practice, there is no need to write down explicitly the canonical form G_(t) ^(g) and the generated sequence before embarking on arithmetic coding. The converting of G_(t) into G_(t) ^(g), constructing of the generated sequence, and encoding of the generated sequence can all be done simultaneously in one pass, assuming that G_(t) has been furnished byour irreducible grammar transform.

FIG. 13 lists the flow diagram of the hierarchical compression algorithm. Steps 1 to 3 above are implemented in Blocks 112 to 119 of FIG. 13.

Although the present invention has been described with reference to preferred embodiments thereof, it will be understood that the invention is not limited to the details thereof. Various modifications and substitutions will occur to those of ordinary skill in the art. All such substitutions are intended to be within the scope of the invention as defined in the appended claims. 

What is claimed is:
 1. A method of sequentially transforming an original data sequence comprising a plurality of symbols into an irreducible grammar from which the original data can be recovered incrementally, wherein the irreducible grammar is represented by a set of production rules which are formed using from a set of variables representing non-overlapping repeated patterns in the data sequence, the method comprising the steps of: (a) parsing a substring from the sequence, wherein the substring is a longest prefix of a string of previously unparsed symbols of the sequence that can be represented by a variable within the set of variables of a previous irreducible grammar other than one of a first variable of the set of variables and a first symbol of the string of the previously unparsed symbols in the sequence; (b) generating an admissible grammar based on the substring and the previous irreducible grammar; (c) applying at least one of a set of reduction rules to the admissible grammar to generate the irreducible grammar; (d) repeating steps (a) through (c) until all of the symbols of the sequence are represented by the irreducible granmmar.
 2. The method according to claim 1, wherein said applying step further comprises the step of implementing a reduction rule comprising the steps of: defining s as a variable of an admissible grammar G that appears once in the range of G; defining a production rule s′→αsβ in which s appears on the right of a group of symbols; defining a production rule s→γ which corresponds to s; and reducing G to an admissible grammar G′ by removing said production rule s→γ from G and replacing said production rule s′→αsβ with the production rule s′→αγβ, the resulting admissible grammar G′ and G representing an equivalent data sequence.
 3. The method according to claim 1, wherein said applying step further comprises the step ofimplementing a reduction rule comprising the steps of: defining G as an admissible grammar possessing a production rule of form s→α₁βα₂βα₃, the length of β being at least 2; defining s′∈S as a variable which is not a G-variable; and reducing G to the grammar G′ by replacing the production rule s→α₁βα₂βα₃ of G with s→α₁s′α₂s′α₃, and appending the production rule s′→β, the resulting grammar G′ having a new variable s′ and representing an equivalent data sequence x as G.
 4. The method according to claim 3, wherein said reducing step is performed in lieu of other ones of said set of production rules when G′ has a non-overlapping repeated pattern of said symbols in the range of G′.
 5. The method according to claim 1, wherein said applying step further comprises the step of implementing a reduction rule comprising the steps of: defining G as an admissible grammar possessing two production rules of form s→α₁βα₂ and s′→α₃βα₄, where β is of length at least two, and either of α₁ and α₂ is not empty, and either α₃ and α₄ is not empty; defining s″∈S as a variable which is not a G-variable; and reducing G to grammar G′ by replacing rule s→α₁βα₂ by s→α₁s″α₂, replacing rule s′→α₃βα₄ by s′→α₃s″α₄, and appending the new rule s″→β.
 6. The method according to claim 1, wherein said applying step further comprises the step of implementing a reduction rule comprising the steps of: defining G as an admissible grammar possessing two production rules of the form s→α₁βα₂ and s′→β, where β is of length at least two, and either of α₁ and α₂ is not empty; and reducing G to the grammar G′ obtained by replacing said production rule s→α₁βα₂ with said production rule s→α₁s′α₂.
 7. The method according to claim 1, wherein said applying step further comprises the step of implementing a reduction rule comprising the steps of: defining G as an admissible grammar in which two variables s and s′ represent the same substring of an A-sequence represented by G; reducing G to a grammar G′ by replacing each appearance of s′ in the range of G by s and deleting a production rule corresponding to s′.
 8. The method according to claim 7, wherein said reducing step further comprises the step of further reducing G′ to the admissible grammar G″ obtained by deleting each said production rules corresponding to variables of G′ that are not involved in the parallel replacement process of G′ if said grammar G′ is not admissible.
 9. The method according to claim 1, wherein a list L₁(α) is allocated to said symbols in said substring, said list L₁(α) comprising vectors (η, s_(m), n), where η∈A U {s₁, . . . , s_(ji)−1}, n is an integer, and having the following properties: (i) γ is the nth element in a symbol array D1(s_(m)), (ii) η is the first element in said symbol array D1(s_(m)) which appears after the nth position and is not equal to −1, (iii) η does not locate in the last position of D1(s₀) when s_(m)=s₀, m=0, (iv) the modified D1(s_(m)) is not equal to γη with the removal of all possible −1 from D1(s_(m)), and (v) n is not equal to the position of the first γ of the pattern in D1(s_(m)) wehen γ=η and when there is a pattern γγγ appearing in the modified D1(s_(m)), and said applying step comprises the step of applying one of a first reduction rule and a second reduction rule if said substring β appears as a first component of some element in said list L₁(α) and if a symbol I(i)=0, said first reduction rule comprising the steps of defining G as an admissible grammar possessing a production rule of form s→α₁βα₂βα₃, the length of β being at least 2, defining s′∈S as a variable which is not a G-variable, and reducing G to the grammar G by replacing the production rule s→α₁βα₂βα₃ of G with s→α₁s′α₂s′α₃, and appending the production rule s′→β, the resulting grammar G′ having a new variable s′ and representing an equivalent data sequence x as G, and said second reduction rule comprising the steps of defining G as an admissible grammar possessing two production rules of form s→α₁βα₂ and s′→α₃βα₄, where β is of length at least two, and either of α₁ and α₂ is not empty, and either α₃ and α₄ is not empty; defining s″∈S as a variable which is not a G-variable, and reducing G to grammar G′ by replacing rule s→α₁βα₂ by s→α₁s″α₂, replacing rule s′→α₃βα₄ by s′→α₃s″α₄, and appending the new rule s″→β.
 10. The method according to claim 9, wherein said applying step further comprises the step of implementing a third reduction rule after said one of a first reduction rule and a second reduction rule if said substring β appears as a first component of some element in said list L₁(α) and if said symbol I(i)=1, said third reduction rule comprising the steps of: defining s as a variable of an admissible grammar G that appears once in the range of G; defining a production rule s′→αsβ in which s appears on the right of a group of symbols; defining a production rule s→γ which corresponds to s; and reducing G to an admissible grammar G′ by removing said production rule s→γ from G and replacing said production rule s′→αsβ with the production rule s′→αγβ, the resulting admissible grammar G′ and G representing an equivalent data sequence.
 11. The method according to claim 9, wherein said generating step comprises the step of appending said substring β to the end of said symbol array D1(s₀) if said substring does not appear as a first component of an element in said list L₁(α).
 12. The method according to claim 11, wherein said appending step further comprises the step of updating said list L₁(α) by inserting an additional vector (η, s_(m), n) where γ locates the nth position of the unappended said symbol array D1(s₀) and is the immediate symbol to the left of the end of the unappended said symbol array D1(s₀).
 13. The method according to claim 12, wherein said appending step further comprises the step of updating a list L₂(α) comprising vectors (η, s_(m), n) having said properties (i), (iii) and (v).
 14. A method of transforming an original data sequence x=x₁x₂ . . . x_(n) comprising a plurality of symbols into a sequence of irreducible grammars {G_(i)}_(i=1) ^(t) from which the original data sequence x can be recovered incrementally, wherein the sequence x=x₁x₂ . . . x_(n) has a finite length n and each grammar G. is represented by a set of production rules {s_(i)→G(s_(i))} formed from a variable set S(j_(i))={s₀, s₁, . . . , s_(j) _(i) ⁻¹} where j_(i) is an integer greater than or equal to 1 and j_(i)=1, the method comprising the steps of: parsing the data sequence x=x₁x₂ . . . x_(n) into t non-overlapping substrings {x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(i−1) ₊₁ . . . x_(n) _(i) }, wherein where i is an integer denoting one of the plurality of non-overlapping substrings, n₁=1, n_(t)=n, and each substring β_(i) is a longest prefix of a string of previously unparsed symbols x_(n) _(i−1) ₊₁ . . . x_(n) of the sequence x that can be represented by a variable s_(j) within the variable set S(j_(i−1))={s₀, s₁, . . . , s_(j) _(i−1) ⁻¹} of the previous irreducible grammar G_(i−1) for 0<j<j_(i−1) or a first symbol x_(n) _(i−1) ₊₁ of the string of the previously unparsed symbols in the sequence; and generating, for each 1<i≦t, an irreducible grammar G_(i) for each x₁ . . . x_(n) _(i) based on each substring β_(i)=x_(n) _(i−1) ₊₁ . . . x_(n) _(i) and on the previous grammar G_(i−1), where G₁ consists of only one production rule {s₀→x₁}.
 15. The method according to claim 14, wherein the step of parsing the sequence x=x₁x₂ . . . x_(n) into a plurality of non-overlapping substrings {x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(i−1) ₊₁ . . . x_(n) _(i) } comprises: determining, for each 1<i≦t, if a prefix of the remaining unparsed sequence x_(n) _(i−1) ₊₁ . . . x_(n) can be represented by a variable s_(j) within the variable set S(j_(i−1))={s₀, s₁, . . . , s_(j) _(i−1) ⁻¹} of the previous irreducible grammar G_(i−1) for 0<j<j_(i−1); setting a current parsed substring β_(i) to be equal to the longest parsed substring x_(n) _(i−1) ₊₁ . . . n_(n) _(i) if x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can be represented by the variable s_(j); and setting the current parsed substring β_(i) to be equal to x_(n) _(i−1) ₊₁ with n_(i)=n_(i−1)+1 if x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can not be represented by the variable s_(j); and wherein the step of determining the irreducible grammar G_(i) for each x₁ . . . x_(n) _(i) based on the substrings β_(i) and the previous grammar G_(i−1) comprises: appending the variable s_(j) to the right end of G_(i−1)(s₀) to generate an admissible grammar G′_(i−1)(s₀) if n_(i)−n_(i−1)>1 and x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can be represented by the variable s_(j); appending x_(n) _(i) to the right end of G_(i−1)(s₀) to generate an admissible grammar G′_(i−1)(s₀) if n_(i)−n_(i−1)=1 or x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can not be represented by a variable s_(j); and applying a set of reduction rules to the admissible grammar G′_(i−1)(s₀) to generate the irreducible grammar G_(i).
 16. A method of sequentially transforming an original data sequence x=x₁x₂ . . . x_(n) comprising a plurality of symbols into a sequence of irreducible grammars {G_(i)}_(i=1) ^(t) from which the original data sequence x can be recovered incrementally, wherein n is the number of symbols in the data sequence x, has a finite length, the method comprising the steps of: parsing a substring β_(i) of unparsed symbols from the data sequence x, wherein the substring β_(i)=x_(n) _(i−1) ₊₁ . . . x_(n) _(i) wherein i is an integer greater between 1 and t inclusive, t is a total number of substrings parsed from the data sequence x and n₁=1; generating an irreducible grammar G_(i) for the symbols x₁ . . . x_(n) _(i) parsed from the data sequence x based on the substring β_(i) and a previous irreducible grammar G_(i−1) corresponding to a previous substring β_(i−1), wherein the irreducible grammar G_(i) is represented by a set of production rules {s_(i)→G(s_(i))} formed from a set of variables S(j_(i))={s_(0, s) ₁, . . . , s_(j) _(i) ⁻¹}, where each variable in the set of variable represents a string of variables from the symbols x₁ . . . x_(n) _(i) , j_(i) is an integer greater than or equal to 1 denoting the number of variables in the variable set S(j_(i)), j₁=1 and G₁ consists of only one production rule {s₀→x₁}; and repeating the above steps until the data sequence x has been parsed into a total of t non-overlapping substrings {x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(i−1) ₊₁ . . . x_(n) _(t) } where n_(t)=n and all of the symbols of the data sequence x are represented by the irreducible grammar G_(t).
 17. The method according to claim 16, wherein the step of parsing a substring β_(i) of symbols from the data sequence x comprises: determining if a prefix of symbols of a remaining unparsed sequence x_(n) _(i−1) ₊₁ . . . x_(n) can be represented by a variable s_(j) within the variable set S(j_(i−1))={s₀, s₁, . . . , s_(j) _(i−1) ⁻¹} of the previous irreducible grammar G_(i−1) for 0<j<j_(i−1); setting the substring β_(i) parsed from the data sequence x to be equal to x_(n) _(i−1) ₊₁ . . . x_(n) _(i) if x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can be represented by the variable s_(j); and setting the substring β_(i) to be equal to x_(n) _(i−1) ₊₁ with n_(i)=n_(i−1)+1 if x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can not be represented by the variable s_(j); and wherein the step of determining the irreducible grammar G_(i) for each x₁ . . . x_(n) _(i) based on the substrings β_(i) and the previous grammar G_(i−1) comprises: appending the variable s_(j) to the right end of G_(i−1)(s₀) to generate an admissible grammar G′_(i−1)(s₀) if n_(i)−n_(i−1)>1 and x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can be represented by the variable s_(j); appending x_(n) _(i) to the right end of G_(i−1)(s₀) to generate an admissible grammar G′_(i−1)(s₀) if n_(i)−n_(i−1)=1 or x_(n) _(i−1) ₊₁ . . . x_(n) _(i) can not be represented by a variable s_(j); and applying a set of reduction rules to the admissible grammar G′_(i−1)(s₀) to generate the irreducible grammar G_(i).
 18. A method of sequentially encoding a data sequence comprising a plurality of symbols, the method comprising the steps of: (a) parsing a current substring of symbols from the data sequence, wherein the current sutbstring is a longest prefix of symbols of previously unparsed symbols of the data sequence representing a non-overlapping repeated pattern in the data sequence or a first symbol of the string of the previously unparsed symbols in the data sequence; (b) encoding the current substring based on a dynamic frequency of the current substring, wherein the dynamic frequency is equal to the number of times the current substring has been parsed from the data sequence divided by a sum of the number of times that each previous substring has been parsed from the data sequence; and (c) repeating steps (a) and (b) until all of the symbols of the sequence are encoded.
 19. The method according to claim 18, wherein said step (a) further comprises generating an irreducible grammar based on the current substring, wherein the irreducible grammar is represented by a set of production rules which are formed using variables of a variable set each representing a non-overlapping repeated pattern in the data sequence.
 20. A method of sequentially transforming a data sequence into a sequence of irreducible grammars and encoding the data sequence based on each of the irreducible grammars, wherein the data sequence comprises a plurality of symbols and the irreducible grammar is represented by a set of production rules which are formed using variables from a variable set, and each variable represents a non-overlapping repeated pattern in the data sequence, the method comprising the steps of: (a) parsing a current substring of symbols from the sequence, wherein the current substring is a longest prefix of a string of previously unparsed symbols of the sequence that can be represented by a variable within the set of variables corresponding a previous irreducible grammar or a first symbol of the string of the previously unparsed symbols in the sequence; (b) encoding the current substring based on a dynamic frequency of the current substring, wherein the dynamic frequency is equal to the number of times the current substring has been parsed from the sequence divided by a sum of the number of times that each previous substring has been parsed from the sequence plus the number of variables; (c) generating an admissible grammar based on the substring and the previous irreducible grammar; (d) applying a set of reduction rules to the admissible grammar to generate the irreducible,grammar; and (e) repeating steps (a) through (d) until all of the symbols of the sequence are parsed and encoded.
 21. A method of sequentially encoding an original data sequence x=x₁x₂ . . . x_(n) comprising a plurality of symbols, the method comprising the steps of: parsing the sequence x=x₁x₂ . . . x_(n) into a plurality of non-overlapping substrings {x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(i−1) ₊₁ . . . x_(n) _(i) }, wherein each substring β_(i)=x_(n) _(i−1) ₊₁ . . . x_(n) _(i) for 1≦i≦t where i is an integer denoting one of the plurality of non-overlapping substrings, t is a total number of substrings parsed from the data sequence x, n₁=1, and n_(t)=n; encoding each substring β_(i) based on a dynamic frequency of the substring, wherein the dynamic frequency is equal to the number of times that the substring β_(i) has been parsed from the data sequence x divided by a sum of the number of times that each previously parsed substrings β₁, . . . , β_(i−1) have been parsed from the sequence (plus the number of variables); and generating, for each 1<i≦t, an irreducible grammar G_(i) for each x₁ . . . x_(n) _(i) based on each substring β_(i)=x_(n) _(i−1) ₊₁ . . . x_(n) _(i) and on the previous grammar G_(i−1), where G₁ consists of only one production rule.
 22. A method of sequentially transforming a data sequence into a sequence of irreducible grammars and encoding the data sequence based on each of the irreducible grammars, wherein the data sequence comprises a plurality of symbols and each of the irreducible grammars is represented by a set of production rules formed from a set of variables each representing a non-overlapping repeated pattern in the data sequence, the method comprising the steps of: (a) parsing a current substring of symbols from the sequence, wherein the current substring is a longest prefix of a string of previously unparsed symbols of the sequence that can be represented by a variable within the set of variables corresponding a previous irreducible grammar or a first symbol of the string of the previously unparsed symbols in the sequence; (b) generating an admissible grammar based on the current substring and the previous irreducible grammar; (c) setting a grammar reduction bit associated with current substring to be equal to one if the admissible grammar is reducible and setting the grammar reduction bit to be equal to zero if the admissible grammar is irreducible; (d) encoding the grammar reduction bit based on the previous and current values of the grammar reduction bit; (e) encoding the current substring based on the previous irreducible grammar, the grammar reduction bit and lists representing consecutive pairs of symbols of the previous irreducible grammar using arithmetic coding; (f) applying a set of reduction rules to the admissible grammar to generate the irreducible grammar; and (g) repeating steps (a) through (f) until all of the symbols of the sequence are parsed and encoded.
 23. A method of encoding a data sequence comprising a plurality of symbols, the method comprising thc steps of: (a) transforming the data sequence into an irreducible grammar from which the original data can be recovered incrementally, wherein the irreducible grammar is represented by a plurality of production rules which are formed from a corresponding set of variables each representing a different non-overlapping repeated pattern in the data sequence; (b) generating a canonical irreducible grammar by converting the irreducible grammar into canonical form; (c) generating a concatenated sequence by concatenating the production rules of the canonical irreducible grammar; and (d) encoding the concatenated sequence using a zero order arithmetic code with a dynamic alphabet.
 24. The method according to claim 23, wherein the step (c) of generating a concatenated sequence further comprises inserting a first symbol at the end of a first production rule.
 25. The method according to claim 24, wherein the step (c) further comprises inserting the first symbol at the end of each production rule having a length greater than two and a second symbol at the beginning of each production rule having a length greater than two.
 26. The method according to claim 23, wherein the step (a) comprises sequentially parsing a plurality of non-overlapping substrings from the sequence and generating an irreducible grammar based on each substring.
 27. A method of encoding an original data sequence x comprising a plurality of symbols, the method comprising the steps of: parsing the data sequence x=x₁x₂ . . . x_(n) into a plurality of non-overlapping substrings {x₁, x₂ . . . x_(n) ₂ , . . . , x_(n) _(i−1) ₊₁ . . . x_(n) _(i) }, wherein each substring β_(i)=x_(n) _(i−1) ₊₁ . . . x_(n) _(i) for 1≦i≦t where i is an integer denoting one of the plurality of non-overlapping substrings, t is a total number of substrings parsed from the data sequence x, n₁=1, and n_(t)=n; and generating a sequence of irreducible grammars {G_(i)}_(i=1) ^(t) for each x₁ . . . x_(n) _(i) based on each substring β_(i)=x_(n) _(i−1) ₊₁ . . . x_(n) _(i) and on the previous grammar G_(i−1), wherein G₁ consists of only one production rule {s₀→x₁)}, each grammar G_(i) is represented by a set of production rules {s_(i)→G(s_(i))} formed from a variable set S(j_(i))={s₀, s₁, . . . , s_(j) _(i) ⁻¹}, j_(i) is an integer greater than or equal to 1 and j₁=1; generating a canonical irreducible grammar G_(t) ^(g) by converting the irreducible grammar G_(t) into canonical form so that the first appearance s_(j) always precedes that of s_(j+1); generating a concatenated sequence of symbols by concatenating the production rules rules {s_(i)→G(s_(i))} of the canonical irreducible grammar; and encoding the concatenated sequence using a zero order arithmetic code with a dynamic alphabet, wherein each symbol β of concatenated sequence is encoded based on a counter c(β) associated with each symbol β which indicates that number of times that each symbol β has occurred in the concatenated sequence. 